Radix sort is one of the linear sorting algorithms for integers. It functions by sorting the input numbers on each digit, for each of the digits in the numbers. However, the process adopted by this sort method is somewhat counterintuitive, in the sense that the numbers are sorted on the least-significant digit first, followed by the second-least significant digit and so on till the most significant digit.
To appreciate Radix Sort, consider the following analogy: Suppose that we wish to sort a deck of 52 playing cards (the different suits can be given suitable values, for example 1 for Diamonds, 2 for Clubs, 3 for Hearts and 4 for Spades). The 'natural' thing to do would be to first sort the cards according to suits, then sort each of the four seperate piles, and finally combine the four in order. This approach, however, has an inherent disadvantage. When each of the piles is being sorted, the other piles have to be kept aside and kept track of. If, instead, we follow the 'counterintuitive' aproach of first sorting the cards by value, this problem is eliminated. After the first step, the four seperate piles are combined in order and then sorted by suit. If a stable sorting algorithm (i.e. one which resolves a tie by keeping the number obtained first in the input as the first in the output) it can be easily seen that correct final results are obtained.
As has been mentioned, the sorting of numbers proceeds by sorting the least significant to most significant digit. For sorting each of these digit groups, a stable sorting algorithm is needed. Also, the elements in this group to be sorted are in the fixed range of 0 to 9. Both of these characteristics point towards the use of Counting Sort as the sorting algorithm of choice for sorting on each digit (If you haven't read the description on Counting Sort already, please do so now).
The time complexity of the algorithm is as follows: Suppose that the n input numbers have maximum k digits. Then the Counting Sort procedure is called a total of k times. Counting Sort is a linear, or O(n) algorithm. So the entire Radix Sort procedure takes O(kn) time. If the numbers are of finite size, the algorithm runs in O(n) asymptotic time.
A graphical representation of the entire procedure has been provided in the attached applet. This program takes 8 randomly generated 3 digit integers as input and sorts them using Radix Sort.
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